Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
9th Edition
ISBN: 9780321962218
Author: Steven J. Leon
Publisher: PEARSON
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Textbook Question
Chapter 3.4, Problem 1E
In Exercise 1 of Section 3.3, indicate whether the given
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1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set
Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k
components, where k is the greatest common divisor of {n, r,s}.
Question 3
over a field K.
In this question, MË(K) denotes the set of n × n matrices
(a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is
equivalent to A-¹? Justify your answer.
(b) Let B be given by
8
B = 0 7 7
0 -7 7
Working over the field F2 with 2 elements, compute the rank of B as an element
of M2(F2).
(c) Let
1
C
-1 1
[4]
[6]
and consider C as an element of M3(Q). Determine the minimal polynomial
mc(x) and hence, or otherwise, show that C can not be diagonalised.
[7]
(d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write
down all the eigenvalues. Show your working.
[8]
R denotes the field of real numbers, Q denotes the field of rationals, and
Fp denotes the field of p elements given by integers modulo p. You may refer to general
results from lectures.
Question 1
For each non-negative integer m, let R[x]m denote the
vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m.
x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent
(a) Let vi = x, V2 =
list in R[x] 3.
(b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4)
is a basis of R[x] 3.
[8]
[6]
(c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a
linear map.
[6]
(d) Write down the matrix for the map ƒ defined in (c) with respect to the basis
(2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3.
[5]
Chapter 3 Solutions
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Ch. 3.1 - Consider the vectors x1=(8,6)T and x2=(4,1)T in R2...Ch. 3.1 - Repeat Exercise 1 for the vectors x1=(2,1)T and...Ch. 3.1 - LetC be the set of complex numbers. Defineaddition...Ch. 3.1 - Show that mn together with the usual additionand...Ch. 3.1 - Show that C[a,b] , together with the usual...Ch. 3.1 - LetP be the set of all polynomials. Show that P,...Ch. 3.1 - Show that the element 0 in a vector space...Ch. 3.1 - Let x, y, and z be vectors in a vector space V....Ch. 3.1 - Let V be a vector space and let xV . Show that (a)...Ch. 3.1 - Lei S be the set of all ordered pairs of real...
Ch. 3.1 - Let V be the set of all ordered pairs of real...Ch. 3.1 - Let R+ denote the set of positive real numbers....Ch. 3.1 - Let R denote the set of real numbers. Define...Ch. 3.1 - Let Z denote the set of ail integers with addition...Ch. 3.1 - LetS denote the set of all infinite sequences of...Ch. 3.1 - We can define a onetoone correspondence between...Ch. 3.2 - Determine whether the following sets form...Ch. 3.2 - Determine whether the following sets form sub...Ch. 3.2 - Determine whether the following are subspaces of...Ch. 3.2 - Determine the null space of each of the following...Ch. 3.2 - Determine whether the following are subspaces of...Ch. 3.2 - Determine whether the following are subspaces of...Ch. 3.2 - Show that Cn[a,b] is a subspace of C[a,b] .Ch. 3.2 - Let A be a fixed vector in nnandletSbethesetof all...Ch. 3.2 - In each of the following determine the subspace of...Ch. 3.2 - LetA be a particular vector in 22 ....Ch. 3.2 - Determine whether the following are spanning...Ch. 3.2 - Which of the sets that follow are spanning sets...Ch. 3.2 - Given x1=(123),x2=(342) x=(266),y=(925) Is...Ch. 3.2 - Let A be a 43 matrixand let b4 . Howmanypossible...Ch. 3.2 - Let A be a 43 matrixandlet c=2a1+a2+a3 (a) If...Ch. 3.2 - Let x1 be a particular solution to a system Ax=b...Ch. 3.2 - Let {x1,x2,...xk} be a spanning set for a vector...Ch. 3.2 - In 22 , let E11=(1000),E12=(0100)...Ch. 3.2 - Prob. 19ECh. 3.2 - Let S be the vector space of infinite...Ch. 3.2 - Prove that if S is a subspace of 1 , then either...Ch. 3.2 - Let Abe an nn matrix. Prove that the...Ch. 3.2 - Let U and V be subspaces of a vector space W.Prove...Ch. 3.2 - Let S be the subspace of 2 spanned by e1 and letT...Ch. 3.2 - Let U and V be subspaces of a vector space W....Ch. 3.2 - Let S, T, and U be subspaces of a vector space V....Ch. 3.3 - Determine whether the following vectors are...Ch. 3.3 - Determine whether the following vectors are...Ch. 3.3 - For each of the sets of vectors in Exercise 2,...Ch. 3.3 - Determine whether the following vectors are...Ch. 3.3 - Let x1,x2,...,xk be linearly independent vectors...Ch. 3.3 - Let x1,x2 , and x3 be linearly independent vectors...Ch. 3.3 - Let x1,x2 , and x3 be linearly independent vectors...Ch. 3.3 - Determine whether the following vectors are...Ch. 3.3 - Prob. 9ECh. 3.3 - Determine whether the vectors cosx,1 , and...Ch. 3.3 - Consider the vectors cos(x+) and sinx in C[,] ....Ch. 3.3 - Given the functions 2x and |x| , show that (a)...Ch. 3.3 - Prove that any finite set of vectors that contains...Ch. 3.3 - Let v1 and v2 be two vectors in a vector space...Ch. 3.3 - Prove that any nonempty subset of a linearly...Ch. 3.3 - Let Abe an mn matrix. Show that if A has linearly...Ch. 3.3 - Let x1,...,xk be linearly independent vectors in n...Ch. 3.3 - Prob. 18ECh. 3.3 - Prob. 19ECh. 3.3 - Let v1,v2,...,vn be linearly independent vectorsin...Ch. 3.4 - In Exercise 1 of Section 3.3, indicate whether the...Ch. 3.4 - In Exercise 2 of Section 3.3, indicate whether the...Ch. 3.4 - Consider the vectors x1=(21),x2=(43),x3=(73) (a)...Ch. 3.4 - Given the vectors x1=(324),x2=(324),x3=(648) what...Ch. 3.4 - Let x1=(213),x2=(314),x3=(264) (a) Show that x1,x2...Ch. 3.4 - In Exercise 2 of Section 3.2, some of the sets...Ch. 3.4 - Find a basis for the subspace S of 4 consisting of...Ch. 3.4 - Given x1=(1,1,1)T and x2=(3,1,4)T : (a) Do x1 and...Ch. 3.4 - Let a1 and a2 be linearly independent vectors in 3...Ch. 3.4 - The vectors x1=(122) , x2=(254) , x3=(132) ,...Ch. 3.4 - Let S be the subspace of P3 consisting of all...Ch. 3.4 - In Exercise 3 of Section 3.2, some of the sets...Ch. 3.4 - In C[,] , find the dimension of the...Ch. 3.4 - In each of the following, find the dimension of...Ch. 3.4 - LetS be the subspace of P3 consisting of all...Ch. 3.4 - In 4 let U be the subspace of all vectors of the...Ch. 3.4 - Is it possible to find a pair of twodimensional...Ch. 3.4 - Show that if U and V are subspaces of n and UV=0 ,...Ch. 3.5 - For each of the following, find the transition...Ch. 3.5 - For each of the ordered bases u1,u2 in Exercise 1,...Ch. 3.5 - Let v1(3,2)T and v2(4,3)T . For each orderedbasis...Ch. 3.5 - Let E=[(5,3)T,(3,2)T] and let x=(1,1)T , y=(1,1)T...Ch. 3.5 - Let u1=(1,1,1)T,u2=(1,2,2)T , and u3=(2,3,4)T (a)...Ch. 3.5 - Let v1=(4,6,7)T,v2=(0,1,1)T , and v3=(0,1,2)T ,...Ch. 3.5 - Given v1=(12) , v2=(23) , S=(351 2) find vectors...Ch. 3.5 - Given v1=(26) , v2=(14) , S=(4121) find vectors u1...Ch. 3.5 - Let [x,1] and [2x1,2x+1] beorderedbasesfor P2 ....Ch. 3.5 - Find the transition matrix representing the...Ch. 3.5 - Let E={u1,...,un} and F={v1,...,vn} be two ordered...Ch. 3.6 - For each of the following matrices, find a basis...Ch. 3.6 - In each of the following, determine the dimension...Ch. 3.6 - Let A=(122314245549367859) (a) Compute the reduced...Ch. 3.6 - For each of the following choices of A and b,...Ch. 3.6 - For each consistent system in Exercise 4,...Ch. 3.6 - How many solutions will the linear system Ax=b...Ch. 3.6 - Let A be a 6n matrix of rank r and let b be a...Ch. 3.6 - Let Abe an mn matrix with mn . Let bRm and suppose...Ch. 3.6 - Let A and B be 65 matrices. If dimN(A)=2 ,what is...Ch. 3.6 - Let A be an mn matrix whose rank is equal to n. If...Ch. 3.6 - Let A be an mn matrix. Prove that rank(A)min(m,n)Ch. 3.6 - Let A and B be row equivalent matrices. (a) Show...Ch. 3.6 - Let A be a 43 matrixandsupposethatthevectors...Ch. 3.6 - Let A be a 44 matrix with reduced row echelonform...Ch. 3.6 - Let A be a 45 matrix and let U be the reduced row...Ch. 3.6 - Let A be a 58 matrix with rank equal to 5 and let...Ch. 3.6 - LetA bea 45 matrix, If a1,a2 , and a4 are...Ch. 3.6 - Let A be a 53 matrix of rank 3 and let {x1,x2,x3}...Ch. 3.6 - Let A be an mnmatrixwithrankequalton.Showthat if...Ch. 3.6 - Prove that a linear system Ax=b is consistent...Ch. 3.6 - LetAandBbemn matrices.Showthat...Ch. 3.6 - Let Abeanmn matrix. (a) Show that if B is a...Ch. 3.6 - Prove Corollary 3.6.4.Ch. 3.6 - Show that if A and B are nn matrices and N(AB)=n...Ch. 3.6 - Prob. 25ECh. 3.6 - Prob. 26ECh. 3.6 - Let x and y be nonzero vectors in m and n ,...Ch. 3.6 - Prob. 28ECh. 3.6 - Let Amn , Bnr , and C=AB . Show that (a) ifA and B...Ch. 3.6 - Prob. 30ECh. 3.6 - An mn matrix A is said to have a right inverse if...Ch. 3.6 - Prove: If A is an mn matrix and the column vectors...Ch. 3.6 - Show that a matrix B has a left inverse if and...Ch. 3.6 - Let B be an nm matrix whose columns arelinearly...Ch. 3.6 - Prob. 35ECh. 3.6 - Prob. 36ECh. 3 - (Change of Basis) Set U=round(20rand(4))10 ,...Ch. 3 - (RankDeficient Matrices) In this exercise we...Ch. 3 - (Column Space arid Reduced Row Echelon Form) Set...Ch. 3 - (Rank1 Updates of Linear Systems) (a) Set...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Prob. 5CTACh. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Prob. 10CTACh. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Prob. 14CTACh. 3 - Prob. 15CTACh. 3 - In 3 , let x1 and x2 be linearly independent...Ch. 3 - For each set that follows determine whether it is...Ch. 3 - Let A=(13134001110022200333) (a) Find a basis for...Ch. 3 - How do the dimensions of the null space and column...Ch. 3 - Answer the following questions and, in each case,...Ch. 3 - Let S be the set of all symmetric 22 matrices with...Ch. 3 - Let A be a 64 matrix of rank 4. (a) What is the...Ch. 3 - Given the vectors x1=(122),x2=(133) ,...Ch. 3 - Let x1,x2 and x3 be linearly independent vectors...Ch. 3 - Let A be a 65 matrix with linearly independent...Ch. 3 - Let {u1,u2} and {v1,v2} be ordered bases for 2 ,...
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